= Uniform forcing of a periodic heat equation
{title2=$u(t)=e^{\alpha t\Delta}u_0+\int_0^tQ(s)\,ds$}
For a spatially constant <PDE source term> $Q(t)$, subtract its time integral from $u$. The result obeys the homogeneous <heat equation> with the same initial condition. This proves the displayed formula and shows that uniform heating changes only the <zero Fourier mode>. In an unnormalised $N_x\times N_y$ <discrete Fourier transform>, a constant increment $q$ adds $N_xN_yq$ to the zero-frequency coefficient.
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